Fractals are geometric shapes that exhibit self-similarity and complex patterns that are repeated at different scales. They can be used to describe irregular shapes found in nature, such as trees, coastlines, and clouds. The document discusses how Benoit Mandelbrot introduced fractals and developed their theory. It provides examples of famous fractals like the Mandelbrot set and Julia set and explains how fractals appear throughout nature and can be modeled using computers. Fractals have applications in fields like mathematics, science, engineering, art and music.

1. Exploring the Wonderful World of
MATHEMATICS
SECRET OF FRACTALS
Dr. Farhana Shaheen
Yanbu University College, KSA

3. WHAT ARE FRACTALS?
• For centuries, mathematicians rejected complex
figures, leaving them under a single description:
“formless”.
• For centuries, geometry was unable to describe
trees, landscapes, clouds, and coastlines. However, in
the late 1970’s a revolution of our perception of the
world was brought by the work of Benoit Mandelbrot
who introduced FRACTALS.

4. Mandelbrot: Inventor of Fractals

5. “Fractua” means
Irregular
• Fractals are geometric figures like circles,
squares, triangles, etc., but having special
properties. They are usually associated with
irregular geometric objects, that look the
same no matter at what scale they are viewed
at.
• A fractal is an object in which the individual
parts are similar to the whole.

6. Fractals exhibit
self-similarity
• Fractals have the property of self-similarity,
generated by iterations, which means that
various copies of an object can be found in the
original object at smaller size scales.
• The detail continues for many magnifications -
- like an endless nesting of Russian dolls within
dolls.

7. Russian dolls within dolls

8. What exactly is a Fractal?
A fractal is a rough or fragmented geometric shape that can
be subdivided into parts, each of which is (at least
approximately) a reduced-size copy of the whole. The core
ideas behind it are of feedback and iteration. The creation of
most fractals involves applying some simple rule to a set of
geometric shapes or numbers and then repeating the process
on the result. This feedback loop can result in very
unexpected results, given the simplicity of the rules followed
for each iteration.
• Fractals have finite area but infinite perimeter.

9. Theory of Fractals
• Mandelbrot introduced and developed the
theory of fractals - figures that were truly able
to describe these shapes. The theory was
continued to be used in a variety of
applications. Fractals’ importance is in areas
ranging from special TV effects to economy
and biology.

10. • The term fractal was coined by Benoit
Mandelbrot in 1975 in his book Fractals: Form,
Chance, and Dimension. In 1979, while studying
the Julia set, Mandelbrot discovered what is now
called the Mandelbrot set and inspired a
generation of mathematicians and computer
programmers in the study of fractals and fractal
geometry.
• Mandelbrot
Set
Mandelbrot’s
discovery

11. The Mandelbrot Set
• Named after Benoit Mandelbrot, The
Mandelbrot set is one of the most famous
fractals in existence. It was born when
Mandelbrot was playing with the simple
quadratic equation z = z2+c.
• In this equation, both z and c are complex
numbers. In other words, the Mandelbrot set
is the set of all complex c such that iteration
z=z2+c does not diverge.

12. The Mandelbrot Set

13. The Julia set

14. The Julia set
• The Julia set is another very famous fractal,
which happens to be very closely related to
the Mandelbrot set. It was named after
Gaston Julia, who studied the iteration of
polynomials and rational functions during the
early twentieth century, making the Julia set
much older than the Mandelbrot set.

15. Difference between the Julia set and the
Mandelbrot set
• The main difference between the Julia set and the
Mandelbrot set is the way in which the function is
iterated. The Mandelbrot set iterates z=z2+c with z
always starting at 0 and varying the c value. The Julia
set iterates z=z2+c for a fixed c value and varying z
values. In other words, the Mandelbrot set is in the
parameter space, or the c-plane, while the Julia set is
in the dynamical space, or the z-plane.

16. The Lorenz Model

17. Lorenz Model
• The Lorenz Model, named after
E. N. Lorenz in 1963, is a model for the
convection of thermal energy. This model was
the very first example of another important
point in chaos and fractals, dissipative
dynamical systems, otherwise known as
strange attractors.

18. A magnification of the Phoenix Set

19. Objects in Nature
Many objects in nature aren’t formed of squares or
triangles, but of more complicated geometric figures.
e.g. trees, ferns, clouds, mountains etc. are shaped
like fractals. Other examples include snow flakes,
crystals, lightning, river networks, cauliflower or
broccoli, and systems of blood vessels and
pulmonary vessels. Coastlines may also be
considered as fractals in nature.

20. Fractals in Nature
• As fractals are patterns that reveal greater
complexity as it is enlarged, they portray the notion
of worlds within worlds.
• Trees and ferns are fractals in nature and can be
modeled on a computer by using a recursive
algorithm. This recursive nature is obvious in these
examples—a branch from a tree or a frond from a
fern is a miniature replica of the whole: not identical,
but similar in nature. The connection between
fractals and leaves are currently being used to
determine how much carbon is contained in trees.

21. Fern leave

22. Fractal Trees

23. Tree Decomposition

24. Fractal Julius Tree
• The rainbow fractal Julius tree below was
created with the help of the computer
program Fractal Imaginator.

25. Fractals in Human Body

26. Examples of Fractals in Nature
• A cauliflower is a perfect example of a fractal
where each element is a perfect recreation of
the whole.

27. A naturally occurring Cauliflower Fractal
• Take a close look at a cauliflower:
Take a closer look at a single floret (break one off
near the base of your cauliflower). It is a mini
cauliflower with its own little florets all arranged
in spirals around a centre.

28. Similarity between fractals and objects
in nature.
One of the largest relationships with real-life is the
similarity between fractals and objects in nature. The
resemblance of many fractals and their natural
counter-parts is so large that it cannot be
overlooked. Mathematical formulas are used to
model self similar natural forms. The pattern is
repeated at a large scale and patterns evolve to
mimic large scale real world objects.

29. Computer-generated Fractal
patterns
• These days computer-generated fractal
patterns are everywhere. From squiggly
designs on computer art posters to
illustrations in most of physics journals,
interest continues to grow among scientists
and, rather surprisingly, artists and designers.

30. Examples of Fractals

32. FRACTALS In Air Balloon

33. Bat Fractal

34. SMOKE FRACTALS

35. FIRE FRACTAL

36. Coastlines Fractal

40. How to make a Koch Snowflake
Fractal

41. Let’s make a Snowflake Fractal
• Fractals are Fun!

42. Pentagram Fractal

43. Animated fractal mountain

44. The Koch Curve Forming of a Fractal

45. Fractal Canopies

46. Dragon Curve

47. Jurassic Park Fractal

48. The Sierpinski Triangle
• Let's make a famous fractal called the Sierpinski
Triangle.
• Step One Draw an equilateral triangle with sides of 2
triangle lengths each.
Connect the midpoints of each side.
• How many equilateral triangles do you now have?

49. The Sierpinski Triangle
• Cut out the triangle in the
center.
• Step Two
• Draw another equilateral triangle with sides of 4
triangle lengths each. Connect the midpoints of the
sides and cut out the triangle in the center as before.

50. The Sierpinski Triangle

51. The Sierpinski Triangle
• Unlike the Koch Snowflake, which is generated with
infinite additions, the Sierpinski triangle is created by
infinite removals. Each triangle is divided into four
smaller, upside down triangles. The center of the
four triangles is removed. As this process is iterated
an infinite number of times, the total area of the set
tends to infinity as the size of each new triangle goes
to zero.

52. Sierpinski Carpet

53. Sierpinski Carpet Cube

54. Hidden Mandarin Fractal Sterling

55. Fractals in Jewelry

56. Fractals in Architecture

57. Medieval Fractal (by Ramiro Perez)

58. Fractals in Lightning

59. Clouds as Fractals

60. Rocky Mountains as Fractals

62. Waterfall Fractals

63. Rivers as Fractals

64. Tsunami

65. Sea Shell Fractal

66. Natural fractal pattern - air displacing a vacuum
formed by pulling two glue-covered acrylic
sheets apart.

67. Fractal Geometry
• Fractal geometry is a new language used to
describe, model and analyze complex forms
found in nature. Chaos science uses this
fractal geometry.
• Fractal geometry and chaos theory are
providing us with a new way to describe the
world.

68. Fractal Geometry
• While the classical Euclidean geometry works
with objects which exist in integer dimensions,
fractal geometry deals with objects in non-
integer dimensions. Euclidean geometry is a
description for lines, ellipses, circles, etc.
Fractal geometry, however, is described in
algorithms -- a set of instructions on how to
create a fractal.

69. Applications of Fractals in Science
• Fractals have a variety of applications in science because its
property of self similarity exists everywhere. They can be used
to model plants, blood vessels, nerves, explosions, clouds,
mountains, turbulence, etc. Fractal geometry models natural
objects more closely than does other geometries.
• Engineers have begun designing and constructing fractals in
order to solve practical engineering problems. Fractals are
also used in computer graphics and even in composing music.

70. Fractals in Music

71. Application of Fractals and Chaos is in Music
• Some music, including that of Bach and
Mozart, can be stripped down so that is
contains as little as 1/64th of its notes and still
retain the essence of the composer. Many
new software applications are and have been
developed which contain chaotic filters,
similar to those which change the speed, or
the pitch of music.

72. Fractals in Music

73. Special Features of Fractals
• A fractal often has the following features:
• It has a fine structure at arbitrarily small scales.
• It is too irregular to be easily described in traditional
Euclidean geometric language.
• It is self-similar (at least approximately or stochastically).
• It has a Hausdorff dimension which is greater than its
topological dimension (although this requirement is not met
by space-filling curves such as the Hilbert curve).
• It has a simple and recursive definition.

74. Application to Biological Analysis
• Fractal geometry also has an application to biological analysis.
Fractal and chaos phenomena specific to non-linear systems
are widely observed in biological systems. A study has
established an analytical method based on fractals and chaos
theory for two patterns: the dendrite pattern of cells during
development in the cerebellum and the firing pattern of
intercellular potential. Variation in the development of the
dendrite stage was evaluated with a fractal dimension. The
order in many ion channels generating the firing pattern was
also evaluated with a fractal dimension, enabling the high
order seen there to be quantized.

76. Fractal Geometry and The Golden Mean

77. Real-Life Relevance And Importance of Fractals
and Fractal Geometry
– Fractals have and are being used in many different
ways. Both artist and scientist are intrigued by the
many values of fractals.
– Fractals are being used in applications ranging
from image compression to finance. We are still
only beginning to realize the full importance and
usefulness of fractal geometry.

78. Fractals in Finance
Fractals in Finance and Risk

79. Fractals in Finance
• Finance played a crucial role in the
development of fractal theory.
• Fractals are used in finance to make
predictions as to the risk involved for
particular stocks.

81. Why does it matter?
• How is the stock market associated with a
fractal? Easily, if one looks at the market price action
taking place on the monthly, weekly, daily and intra
day charts where you will see the structure has a
similar appearance. Followers of this approach have
determined that market prices are highly random but
with a trend. They claim that stock market success
will happen only by following the trend.

82. Applications of Fractals
– One of the most useful applications of fractals and
fractal geometry is in image compression. It is also
one of the more controversial ideas. The basic
concept behind fractal image compression is to
take an image and express it as an iterated system
of functions. The image can be quickly displayed,
and at any magnification with infinite levels of
fractal detail. The largest problem behind this idea
is deriving the system of functions which describe
an image.

83. Fractals in Film Industry
One of the more trivial applications of fractals is their
visual effect. Not only do fractals have a stunning
aesthetic value, that is, they are remarkably pleasing
to the eye, but they also have a way to trick the
mind. Fractals have been used commercially in the
film industry, in films such as Star Wars and Star Trek.
Fractal images are used as an alternative to costly
elaborate sets to produce fantasy landscapes.

84. Other Applications of Fractals
• As described above, random fractals can be used to describe
many highly irregular real-world objects. Other applications of
fractals include:
• Classification of histopathology slides in medicine
• Fractal landscape or Coastline complexity
• Enzyme/enzymology (Michaelis-Menten kinetics)
• Generation of new music
• Signal and image compression
• Creation of digital photographic enlargements
• Seismology
• Fractal in soil mechanics

85. • Computer and video game design, especially computer
graphics for organic environments and as part of procedural
generation
• Fractography and fracture mechanics
• Fractal antennas – Small size antennas using fractal shapes
• Small angle scattering theory of fractally rough systems
• T-shirts and other fashion
• Generation of patterns for camouflage, such as MARPAT
• Digital sundial
• Technical analysis of price series (see Elliott wave principle)

86. Applications of Fractals in Computer
Science
• fractal techniques for data analysis
• fractals and databases, data mining
• visualization and physical models
• automatic object classification
• fractal and multi-fractal texture
characterization
• shape generation, rendering techniques and
image synthesis
• 2D, 3D fractal interpolation
• image denoising and restoration
• image indexing, thumbnail images

87. • fractal still image and video compression, wavelet
and fractal transforms, benchmarking, hardware
• watermarking, comparison with other techniques
• biomedical applications
• engineering (mechanical & materials, automotive)
• fractal and compilers, VLSI design
• internet traffic characterization and modeling
• non classical applications

88. • Dear Students and Colleagues,
This is not THE END…
This is just the beginning…
… to start exploring…
… the Wonderful World of Mathematics
Thank You
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